Hi I uploaded them, as old maps show there are quite a few "Old Shafts" or "Air Shafts" littered around, I have studied the old maps and cannot find 1st Pit ? But as mentioned the 2nd Pit was located in the car park (on the left) as you approach Beamish Museum, Shepherd and Shepherdess Public House on the right.
It is believed that this Air shaft was for the workings toward The Mahogany Drift Mine which is situated in the Museum Pit village, There was also a few shafts and later a Fan to the East and a number of shafts behind the 2nd Pit near the road behind the museum.
I'm unsure if there were older than the 2nd Pit or just abandoned (remember the age of these mines Air shafts were necessary due to the primitive ventilation) but all these shafts indicate they were working Noth toward the Mahogany drift.
The Mary Shaft was an Air shaft and Pit that turned into a Mine (outliving the 2nd Pit, although that mine sat abandoned until the 1970s when The Museum saved it and rebuilt it in the Museum).
Hope this helps, I have uploaded The Beamish Mary, with some maps to follow.